The compound Poisson risk model under a mixed dividend strategy

Applied Mathematics and Computation 315 (2017) 1–12

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

The compound Poisson risk model under a mixed dividend strategy Zhimin Zhang a,∗, Xiao Han b a b

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR China College of Science, PLA University of Science and Technology, Nanjing 211101, PR China

a r t i c l e

i n f o

a b s t r a c t

Keywords: Compound Poisson model Mixed dividend strategy Expected discounted dividends Expected discounted penalty function Ruin

In this paper, we consider a compound Poisson model under a mixed dividend strategy. The mixed dividend strategy is a combination of threshold dividend strategy and periodic dividend strategy. Given a positive threshold level b > 0, whenever the surplus process attains the level b, dividends will be paid off continuously at a rate α > 0. Furthermore, given ∞ a sequence of dividend decision times Z j , whenever the observed surplus level at Zj is j=1

larger than b, the excess value will also be paid off as dividend. We study the expected discounted dividend payments before ruin and the Gerber–Shiu expected discounted penalty function. Some numerical examples are also presented. © 2017 Elsevier Inc. All rights reserved.

1. Introduction The classical risk model of an insurance company is defined as

Ut = u + ct − St , t ≥ 0,

Nt

(1.1)

where u ≥ 0 is the initial surplus, c > 0 is the constant premium rate, and St = n=1 Yn is the aggregate claims process up to time t. The claim number process {Nt : t ≥ 0} is a homogeneous Poisson process with intensity λ > 0. The individual claim amounts Y1 , Y2 , . . . form a sequence of positive i.i.d. random variables with common density function fY and Laplace trans∞ form  fY (s ) = 0 e−sy fY (y )dy. The classical risk model has been well studied and a lot of extensions have been proposed. One of the extensions is the dividend modification, where a fraction of the surplus will be paid as dividends to shareholders whenever the insurance company is under some net profit conditions. One popular dividend strategy is the threshold dividend strategy, which is defined as follows. Given a threshold level b > 0, whenever the surplus process attains the level b, dividends will be paid at rate α (0 < α < c). The modified surplus process under the threshold dividend strategy, denoted   by U¯ t , can be described as follows: t≥0

dU¯ t =



cdt − dSt ,

0 ≤ U¯ t ≤ b,

(c − α )dt − dSt ,

U¯ t > b.

(1.2)

Note that there are two dynamic parts on the right hand side of (1.2). The first part cdt − dSt means that surplus process evolves linearly at rate c between successive claim arrival times when it is below the threshold level b, while the second ∗

Corresponding author. E-mail addresses: [email protected], [email protected] (Z. Zhang).

http://dx.doi.org/10.1016/j.amc.2017.07.048 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

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Z. Zhang, X. Han / Applied Mathematics and Computation 315 (2017) 1–12

dynamic part (c − α )dt − dSt means that surplus process evolves linearly at rate c − α between successive claim arrival times when it is above the threshold level b. The second dynamic evolution rate is less than the first one due to that parts of premiums income are paid off as dividends when the surplus process is above the level b. For the study of the threshold dividend strategy, see e.g. [1,13] and [19]. The periodic dividend strategy was first proposed by Albrecher et al. [1], where decisions on paying dividends are only made at a sequence of discrete time points. Since then, a lot of contributions have been made to this strategy. Avanzi et al. [4] study the Laplace transform of the ruin time and the expected discounted dividends until ruin in a dual risk model under the periodic dividend strategy, where the inter-decision times follow Erlang distribution. Choi and Cheung [7] study the expected discounted dividends in the Crame´ r –Lundberg risk model with more frequent ruin monitoring than dividend decisions. Zhang [21] and Zhang and Liu [25] study the Gerber–Shiu function and the expected discounted dividend payments in a perturbed compound Poisson model under the periodic dividend strategy. Zhang and Cheung [22] consider an Erlangized dividend barrier strategy in a more general risk model, namely, the Markov additive risk process. Zhang et al. [23] propose a Lévy risk model with periodic tax payments, where the inter-decision times follow exponential distribution. Zhang et al. [24] consider a compound Poisson risk model with capital injections, where the inter-decision times are assumed to follow Erlang distribution. We remark that the dynamic process described below has also existed in the realm of complex system modeling (e.g. [5,6,8]) and evolutionary games (e.g. [14,20]). In this paper, we propose a mixed dividend strategy, which is a combination of the threshold dividend strategy and the  ∞ periodic dividend strategy. We modify the process U¯ as follows. Let Z j (0 < Z1 < Z2 < · · · ) be a sequence of periodic j=1

dividend decision times. Whenever the observed surplus level at Zj is larger than b, the excess value will also be paid off as   dividend. In order to give the mathematical descriptions of the modified surplus process U b = Utb under mixed dividend



strategy, we introduce auxiliary processes U¯ j = U¯ j,t as follows:

U¯ j,t =

⎧ ⎨U¯t ,



t≥0

t≥0

for j = 1, 2, . . . . The dynamics of Ub and U¯ j can be jointly described

j = 1; t ≥ 0,







⎩U b +  t c − α I U¯ j,s > b ds − Nt n=NZ Z j−1 j−1

j−1 +1

Yn ,

j = 2, 3, . . . ; t ≥ Z j−1 ,

and for j = 1, 2, . . . ,

Utb =

⎧ ⎨U¯ j,t ,

Z j−1 < t < Z j ,





⎩min U¯ , b , j,Z j

(1.3) t = Z j.

We plot a sample path of Ub in Fig. 1. Set Z0 = 0 (but it is not a dividend decision time) in the above definition, and therefore U0b = u even if U0b = u > b. Assuming that the event of ruin is monitored continuously, the ruin time of Ub is defined by

  τ b = inf t > 0 : Utb < 0

with the convention inf ∅ = ∞. In this paper, we are interested in the Gerber–Shiu expected discounted penalty function that is defined as

      b φ (u; b) = E e−δτ w Uτbb − , Uτbb  I τ b < ∞ U0b = u , u ≥ 0,

where δ ≥ 0 is the Laplace transform argument of the ruin time or the force of interest, and w : [0, ∞ ) × (0, ∞ ) → [0, ∞ ) is a nonnegative penalty function of the surplus before ruin and the deficit at ruin. The Gerber–Shiu function was first proposed by Gerber and Shiu [9]. Since then, it has become a standard tool for studying ruin related quantities, and a lot of contributions to it have been made by actuarial researchers. For its study in the risk model with periodic strategic decisions, we refer the interested readers to [3,15,21–26]. Another quantity of interest is the expected discounted dividend payments before ruin. Note that there are following two types of dividend payments, • type 1: continuous dividend payments at rate α ; • type 2: periodic dividend payments at dividend decision times Zj , j = 1, 2, . . . . Hence, the total discounted dividend payments up to ruin is given by

Dbδ =

J   j=1

t∧τ b

Z j−1

e −δ s

J     e−δ j U¯ j,Z j − b , α I U¯ j,s > b ds + +



where x+ = max(x, 0 ), J = max j : Z j ≤ τ

   V (u; b) = E Dbδ U0b = u , u ≥ 0.

 b

j=1

. Then the expected discounted dividend payments before ruin is defined by

Z. Zhang, X. Han / Applied Mathematics and Computation 315 (2017) 1–12

3

dividend(type1) Ub t

dividend(type1) dividend(type2) dividend(type1)

b

u

Z

5

0

Z2

Z1

Z3

t

Z4

Fig. 1. A sample path of the surplus process Ub .

Throughout this paper, we study φ (u; b) and V(u; b) under the assumption that the inter-dividend-decision times are exponentially distributed. More precisely, let T1 = Z1 be the first dividend decision time and for j ≥ 2, let T j = Z j − Z j−1 be the time interval between the ( j − 1)th and jth decision times. It is assumed that

 ∞ Tj

j=1

are i.i.d. with common density

function

fT (t ) = γ e−γ t ,

γ , t > 0.

The reminder of this paper is organized as follows. In Section 2, we derive integro-differential equations for V(u; b) and discuss the solution procedure. Similarly, φ (u; b) is studied in Section 3. In Section 4, we give various numerical examples to illustrate our theoretical results. Finally, some conclusion remarks are given in Section 5. 2. Analysis of V(u; b) In this section, we study the expected discounted dividend payments prior to ruin. Since V(u; b) behaves differently according to 0 ≤ u ≤ b and u > b, we put



V ( u; b ) =

V1 (u ),

0 ≤ u ≤ b,

V2 (u ),

u > b.

First, we study V1 (u). For 0 ≤ u ≤ b, by conditioning on the competition between the first claim arrival time and the first dividend decision time, we obtain

V1 (u ) =



b−u c

0



+  +  +  +

λe−(λ+γ +δ )t

∞ b−u c

∞ b−u c

λe−(λ+γ )t

∞ b−u c



u+ct 0

λe−(λ+γ +δ )t

b−u c

0





V (u + ct − y; b) fY (y )dydt

(

b+(c−α ) t− b−u c 0

t b−u c

)



γ e−(λ+γ )t

t b−u c

b−u c



 − y; b fY (y )dydt

e−δ s (c − α )dsdt

γ e−(λ+γ +δ )t V (u + ct; b)dt + 



V b + (c − α ) t −



e−δ s (c − α )dsdt

∞ b−u c

   b−u γ e−(λ+γ +δ )t V (b; b) + (c − α ) t − dt c

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Z. Zhang, X. Han / Applied Mathematics and Computation 315 (2017) 1–12

 =

b−u c

0



∞

+

λe

b−u c

 +

λe−(λ+γ +δ )t

b−u c

0



u+ct 0

−(λ+γ +δ )t



V (u + ct − y; b) fY (y )dydt

(

b+(c−α ) t− b−u c

)

0





b−u V b + (c − α ) t − c



 − y; b fY (y )dydt

γ e−(λ+γ +δ )t V (u + ct; b)dt

(c − α )(λ + 2γ + δ ) + γ (λ + γ + δ )V (b; b) −(λ+γ +δ ) b−u c . e ( λ + γ + δ )2

+

(2.1)

Applying some changes of variables in (2.1) we obtain



λ

V1 (u ) =

c

b

e−(λ+δ +γ )

u

c−α  b c

∞

V (s − y; b) fY (y )dyds

e−(λ+δ +γ )( c−α + s−b

b−u c

b

γ

+

s

0



λ

+



s−u c

e−(λ+γ +δ )

b−u c

u

)V (s − y; b) fY (y )dyds

V (s; b)ds +

(c − α )(λ + 2γ + δ ) + γ (λ + γ + δ )V (b; b) −(λ+γ +δ ) b−u c . e ( λ + γ + δ )2

Differentiating both sides of the above equation gives

d V1 (u ) = du

λ+δ c

λ

V1 (u ) −



c

u 0

V1 (u − y ) fY (y )dy, 0 ≤ u ≤ b.

(2.2)

The solution of IDE (2.2) is given by

V1 (u ) = kv(u ),

(2.3)

where k is an unknown constant to be determined, and v(u ) with boundary condition v(0 ) = 1 is the solution to the following homogeneous IDE,

λ+δ

d v (u ) = du

v (u ) −

c



λ c

u

0

v(u − y ) fY (y )dy, u ≥ 0.

(2.4)

Remark 1. The solution of (2.4) can be determined by Laplace transform. Define  v= forms on both sides of (2.4) gives

 v=

1 s − λ+c δ + λc  f (s )

∞ 0

e−su v(u )du. Taking Laplace trans-

.

When  f (s ) belongs to the rational family (for example, fY (y) is a combination of exponentials or Erlang), the explicit expression for v(u ) can be obtained by Laplace inversion. Next, we consider the case u > b. Again, by conditioning on the time and amount of the first claim we obtain

V2 (u ) =



∞ 0



+  +

λe−(λ+γ +δ )t ∞

0 ∞ 0

λ

=

λe−(λ+γ )t



0



t 0

V (u + (c − α )t − y; b) fY (y )dydt

e−δ s (c − α )dsdt

∞

e−(λ+γ +δ ) c−α s−u



u

s 0



∞ 0

γ e−(λ+γ )t



t 0

e−δ s (c − α )dsdt

V (s − y; b) fY (y )dyds

γ λ + 2γ + δ (V (b; b) + u − b) + ( c − α ). λ+γ +δ ( λ + γ + δ )2

Applying the operator

d − du

u+(c−α )t

γ e−(λ+γ +δ )t (V (b; b) + u + (c − α )t − b)dt +

c−α +





d du

−



(2.5)

λ+ γ + δ to (2.5) gives c−α

λ+γ +δ λ V2 (u ) = − c−α c−α



0

u

V (u − y; b) fY (y )dy −

γ

c−α

(V (b; b) + u − b) − 1.

(2.6)

To solve (2.6), we introduce the Dickson–Hipp operator Ts , that is defined as: for any integrable function f on [0, ∞),

Ts f ( x ) =



x

∞

e−s(y−x ) f (y )dy =



0

∞

e−sy f (x + y )dy, Re(s ) ≥ 0.

Z. Zhang, X. Han / Applied Mathematics and Computation 315 (2017) 1–12

5

For the properties of Dickson–Hipp operator, we refer the interested authors to [8] and [12]. Now multiplying both sides of (2.6) by e−s(u−b) and performing integration by parts, we obtain



∞



e−s(u−b)

b

 λ+δ+γ λ+δ+γ V2 (u )du = sTsV2 (b) − V2 (b) − TsV2 (b). c−α c−α

d − du

By changing the order of integrals, we have



∞

e−s(u−b)



b

u 0

V (u − y; b) fY (y )dydu =  fY (s )TsV2 (b) +

 0

b

V1 (y )Ts fY (b − y )dy.

Hence, (2.6) becomes

λ+δ+γ TsV2 (b) c−α    b −λ  λ γ V (b; b) 1 γ 1 = fY (s )TsV2 (b) − V1 (y )Ts fY (b − y )dy − 1 + − , c−α c−α 0 c−α s c − α s2

sTsV2 (b) − V2 (b) −

which yields



 λ+δ+γ λ  s− + fY (s ) TsV2 (b) c−α c−α



 γ V (b; b) 1 γ 1 = V2 (b) − V1 (y )Ts fY (b − y )dy − 1 + − c−α 0 c−α s c − α s2    b λ γ v (b ) 1 γ 1 = kv ( b ) − k v(y )Ts fY (b − y )dy − 1 + k − . c−α 0 c−α s c − α s2 

λ

b

(2.7)

Since the following equation

s−

λ+δ+γ λ  + fY (s ) = 0 c−α c−α

has a unique positive root ρ (see e.g. [9]), then the left hand side of (2.7) yields

λ+δ+γ λ  + fY (s ) c−α c−α     λ+δ+γ λ  λ+δ+γ λ  = s− + fY (s ) − ρ − + fY (ρ ) c−α c−α c−α c−α λ  = (s − ρ ) − fY (ρ ) −  fY (s ) c−α   λ = (s − ρ ) 1 − Ts Tρ fY (0 ) . c−α s−

On the other hand, setting s = ρ in (2.7) gives

kv ( b ) =



λ

c−α

b

k 0

 1 γ γ 1 v(y )Tρ fY (b − y )dy + 1 + kv ( b ) + . c−α ρ c − α ρ2

Hence, the right hand side of (2.7) equals to

  1 1    1 γ γ 1 v(y ) Tρ fY (b − y ) − Ts fY (b − y ) dy + 1 + kv ( b ) − + − c−α 0 c−α ρ s c − α ρ2 s2    b   γ v (b ) 1 γ s+ρ λ = ( s − ρ )k v(y )Ts Tρ fY (b − y )dy + + (s − ρ ) + c−α 0 c − α sρ c − α s2 ρ 2 sρ  = (s − ρ )  h0 (s ) + k h1 ( s ) , λ



b

k

where  h0 (s ) and  h1 (s ) are Laplace transforms of the functions



γ



γ + u, ρ (c − α ) ρ 2 (c − α ) ρ  b λ γ v (b ) h1 ( u ) = v(y )Tρ fY (u + b − y )dy + . c−α 0 c−α ρ h0 ( u ) =

+

1

(2.8)

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Z. Zhang, X. Han / Applied Mathematics and Computation 315 (2017) 1–12

Substituting (2.8) back into (2.7) gives



1−



λ

c−α

Ts Tρ fY (0 ) TsV2 (b) =  h0 (s ) + k h1 ( s ),

or

λ

TsV2 (b) =

c−α

Ts Tρ fY (0 ).TsV2 (b) +  h0 (s ) + k h1 ( s ).

By Laplace inversion we obtain



V2 (u + b) =

u 0

V2 (u + b − y )g(y )dy + h0 (u ) + kh1 (u ),

(2.9)

where

λ

g( y ) =

c−α

Tρ fY (y ).

Under the regular condition c − α > λEY, g(y) is a defective density in the sense that  ∗n Let S(y ) = ∞ n=1 g (y ). Then the solution of (2.9) can be expressed as

V2 (u + b) = h0 (u ) + kh1 (u ) +



u 0

∞ 0

g(y )dy < 1 (see e.g. [9]).

(h0 (u − y ) + kh1 (u − y ) )S(y )dy

π0 (u ) + kπ1 (u ), u ≥ 0,

= where

π0 ( u ) = h 0 ( u ) + π1 ( u ) = h 1 ( u ) +



u

0



u 0

(2.10)

h0 (u − y )S(y )dy, h1 (u − y )S(y )dy.

In order to determine k, we use the continuity condition of V(u; b) at u = b. Then we have

kv(b) = π0 (0 ) + kπ1 (0 ), which gives

k=

π0 ( 0 ) . v ( b ) − π1 ( 0 )

Finally, we have

⎧ π ( 0 )v ( u ) ⎨ v(b0 )−π1 (0) ,

V ( u; b ) =

0 ≤ u ≤ b,

⎩π (u − b) + 0

π0 (0 )π1 (u−b) v ( b ) − π1 ( 0 ) ,

u > b.

3. The expected discounted penalty function In this section, we study the expected discounted penalty function. For convenience, we put

 φ1 ( u ) ,

φ ( u; b ) =

0 ≤ u ≤ b,

φ2 ( u ) ,

u > b.

First, we study φ 1 (u). For 0 ≤ u ≤ b, by conditioning on the competition between the first claim arrival time and the first dividend decision time, and considering the size of the first claim, we obtain

φ1 ( u )  =

b−u c

0



+  +

λe−(λ+δ+γ )t b−u c

0 ∞ b−u c



u+ct 0

λe−(λ+δ+γ )t

λe

−(λ+δ +γ )t



 0

φ (u + ct − y; b) fY (y )dydt

∞

u+ct

w(u + ct, y − u − ct ) fY (y )dydt

(

b+(c−α ) t− b−u c

)

 φ



b−u b + (c − α ) t − c



 − y; b fY (y )dydt

Z. Zhang, X. Han / Applied Mathematics and Computation 315 (2017) 1–12



∞

+

λe−(λ+δ+γ )t

b−u c







∞ b+(c−α )(

t− b−u c



w b + (c − α )(t −

)



b−u c

b−u b−u ), y − b − (c − α ) t − c c

7

 fY (y )dydt

∞

γ e−(λ+δ+γ )t φ (u + ct; b)dt + γ e−(λ+δ+γ )t φ (b; b)dt b−u 0 c  s    ∞ λ b −(λ+δ+γ ) s−uc = e φ (s − y; b) fY (y )dy + w(s, y − s )dy ds c 0 0 s  s   ∞  ∞ s−b b−u λ + e−(λ+δ +γ )( c−α + c ) φ (s − y; b) fY (y )dy + w(s, y − s ) fY (y )dy ds c−α b 0 s +

+



γ c

b

e−(λ+δ +γ )

s−u c

u

φ (s; b)ds +

γ b−u φ (b; b)e−(λ+δ+γ ) c . λ+δ+γ

Differentiating both sides of (3.1) w.r.t. u gives

λ+δ+γ

d φ1 ( u ) = du

c

λ+δ

= where

ω (u ) =



∞ u

φ1 ( u ) −

φ1 ( u ) −

c

λ c



λ



c u

u

0



φ1 (u − y ) fY (y )dy + ω (u ) −

(3.1)

γ c

φ ( u; b )

λ φ1 (u − y ) fY (y )dy − ω (u ),

(3.2)

c

0

w(u, y − u ) fY (y )dy.

By [9] we know that φ (u; ∞) satisfies the following IDE:

λ+δ

φ ( u; ∞ ) =

c

φ ( u; ∞ ) −

λ c



u 0

λ φ (u − y; ∞ ) fY (y )dy − ω (u ), u ≥ 0. c

Hence, by the general theory on differential equation, we have

φ1 ( u ) = φ ( u ; ∞ ) + l · v ( u ) ,

(3.3)

where l is an unknown constant and v(u ) is defined as in Section 2. Next, we study φ 2 (u). By conditioning on the competition between the first claim arrival time and the first dividend decision time, and considering the size of the first claim, we obtain

φ2 ( u ) =



∞ 0



+  + =

λe−(λ+δ+γ )t ∞

0 ∞ 0

λ

 0

λe−(λ+δ+γ )t

u+(c−α )t



φ (u + (c − α )t − y; b) fY (y )dydt

∞ u+(c−α )t

w(u + (c − α )t, y − u − (c − α )t ) fY (y )dydt

γ e−(λ+δ+γ )t φ (b; b)dt  s  ∞ e−(λ+γ +δ ) c−α s−u

c−α

u

0

φ (s − y; b) fY (y )dy +



∞ s

 w(s, y − s ) fY (y )dy ds

γ φ (b; b) + . λ+γ +δ

(3.4)

Differentiating both sides of (3.4) w.r.t. u gives



d − du

  u λ+δ+γ λ λ γ φ (b; b) φ (u − y; b) fY (y )dy − . φ2 ( u ) = − ω (u ) − c−α c−α 0 c−α c−α

(3.5)

Using the same arguments as in Section 2, it is easy to obtain



 λ+γ +δ λ  s− + fY (s ) Ts φ2 (b) c−α c−α

= φ2 ( b ) −



λ

c−α

b 0

φ1 (y )Ts fY (b − y )dy −

λ

c−α

Ts ω ( b ) −

γ φ (b; b) 1 . c−α s

Setting s = ρ in (3.6) gives

φ2 ( b ) =

λ

c−α



0

b

φ1 (y )Tρ fY (b − y )dy +

λ

c−α

Tρ ω ( b ) +

γ φ (b; b) 1 , c−α ρ

(3.6)

8

Z. Zhang, X. Han / Applied Mathematics and Computation 315 (2017) 1–12

which immediately results in





λ

φ2 ( b ) −

c−α

b 0

λ

φ1 (y )Ts fY (b − y )dy −

c−α

γ φ (b; b) 1 c−α s

φ1 (y ) Tρ fY (b − y ) − Ts fY (b − y ) dy c−α 0 γ φ (b; b)  1 1  λ  + Tρ ω ( b ) − Ts ω ( b ) + − c−α c−α ρ s    b λ γ φ (b; b) 1 λ = (s − ρ ) φ1 (y )Ts Tρ fY (b − y )dy + Ts Tρ ω ( b ) + . c−α 0 c−α c − α sρ λ

=



Ts ω ( b ) −

b

Then (3.6) becomes



1−

λ

c−α



λ

Ts Tρ fY (0 ) Ts φ2 (b) =



c−α

b 0

φ1 (y )Ts Tρ fY (b − y )dy +

λ

c−α

(3.7)

Ts Tρ ω ( b ) +

γ φ (b; b) 1 , c − α sρ

which yields

T s φ2 ( b ) = =

λ

c−α

λ

λ

c−α

λ

Ts Tρ ω ( b ) +

γ

c−α

γ 1 φ (b; ∞ ) + l c − α sρ λ





λ

c−α

λ



c−α

b 0 b 0

φ1 (y )Ts Tρ fY (b − y )dy +

λ



λ

c−α  b

c−α

0

b 0

λ

c−α

Ts Tρ ω ( b ) +

(φ (y; ∞ ) + l · v(y ) )Ts Tρ fY (b − y )dy

(φ (b; ∞ ) + l · v(b) )

Ts Tρ fY (0 ) · Ts φ2 (b) +

c−α +

=

Ts Tρ fY (0 ) · Ts φ2 (b) +

c−α +

=

Ts Tρ fY (0 ) · Ts φ2 (b) +

1 sρ

φ (y; ∞ )Ts Tρ fY (b − y )dy +

v(y )Ts Tρ fY (b − y )dy +

λ

c−α

γ v (b ) c − α sρ



Ts Tρ ω ( b )

 0 (s ) + l ·  Ts Tρ fY (0 ) · Ts φ2 (b) + m h1 ( s ),

c−α

0 (s ) is Laplace transforms of the function where  h1 (s ) is defined as in Section 2, and m

m0 ( u ) =



λ

c−α

0

b

φ (y; ∞ )Tρ fY (u + b − y )dy +

λ

c−α

Tρ ω ( u + b ) +

γ

c−α

1

ρ

φ (b; ∞ ).

By Laplace inversion we obtain

φ2 ( u + b ) =

 0

u

φ2 (u + b − y )g(y )dy + m0 (u ) + l · h1 (u )

= m0 ( u ) + l · h1 ( u ) + = m0 ( u ) + =

 0



u

(m0 (u − y ) + l · h1 (u − y ))S(y )dy    u u m0 (u − y )S(y )dy + l · h1 (u ) + h1 (u − y )S(y )dy 0

π2 ( u ) + l · π1 ( u ) ,

where

π2 ( u ) = m 0 ( u ) +

 0

u

m0 (u − y )S(y )dy.

and π 1 (u) is defined as in Section 2. By the continuity condition of φ (u; b) at u = b, we have

φ (b; ∞ ) + l · v(b) = π2 (0 ) + l · π1 (0 ), which gives

l=

π2 (0 ) − φ (b; ∞ ) . v ( b ) − π1 ( 0 )

Finally, we have

0

γ φ (b; b) 1 c − α sρ

Z. Zhang, X. Han / Applied Mathematics and Computation 315 (2017) 1–12

a

b

9

c

40

40

35

35

35

30

30

30

V(u;b)

V(u;b)

V(u;b)

25

25

25

20 20

20 b=3 15

b=3

2

4

6

8

b=8

b=8 10

0

b=5

b=5

b=8 10

b=3

15

15

b=5

10

10 0

1

2

3

u

4

5 u

6

7

8

9

10

0

2

4

6

8

Fig. 2. The expected discounted dividends before ruin as a function of u for b = 3, 5, 8. (a) fY (y ) = 3e− 2 y − 3e−3y ; (b) fY (y ) = e−y ; (c) fY (y ) = 4 −2y e . 3 3

φ ( u; b ) =

⎧ ⎨ φ ( u; ∞ ) +

π2 (0 )−φ (b;∞ ) v ( b ) − π1 ( 0 ) · v ( u ) ,

⎩ π (u − b ) + 2

π2 (0 )−φ (b;∞ ) v ( b ) − π1 ( 0 ) · π 1 ( u − b ) ,

10

u

1 − 12 y e 6

+

0 ≤ u ≤ b, u > b.

Hence, the expected discounted penalty function is completely determined. 4. Numerical examples In this section, we shall apply the theoretical results derived in Sections 2 and 3 to provide some numerical examples on the expected discounted dividend payments before ruin and the Gerber–Shiu expected discounted penalty function. Throughout this section, we set δ = 0.05, γ = 0.5, λ = 1, c = 2, α = 0.5. For the claim size density function, we consider the following three cases: (1) a sum of two exponentials with mean 1/3 and 2/3; (2) an exponential distribution with mean 1; (3) a mixture of two exponentials: one exponential with mean 2 (mixing probability 1/3) and one exponential with mean 1/2 (mixing probability 2/3). Note that the above probability density functions all belong to the combinations of exponentials, and the corresponding expressions are given as follows, 3

(1) fY (y ) = 3e− 2 y − 3e−3y ; (2) fY (y ) = e−y ; (3) fY (y ) =

1 − 12 y 6e

+ 43 e−2y .

Moreover, we find that the common mean of these three distributions is 1, while the variances are 0.56, 1 and 2, respectively. The above examples have also been considered by Albrecher et al. [2]. We start with the analysis of the expected discounted dividend payments before ruin. In Figs. 2 and 3, we depict the behavior of V(u; b) as functions of u and b, respectively. First, for fixed threshold levels b = 3, 5, 8, we find from Fig. 2 that V(u; b) is an increasing function of the initial surplus level u. Intuitively, when the initial surplus increases, there are two effects to the amount of dividend payments. On one hand, with a larger initial surplus the surplus process is more likely to be above the threshold level b. On the other hand, a larger initial surplus also implies that the surplus process survives longer and hence there could be potentially more dividend payments at later times. Furthermore, Fig. 2 does not imply that V(u; b) is a monotone function of b. To find the behavior of V(u; b) as a function of b, we plot curves for V(u; b) in Fig. 3 for initial surplus levels u = 3, 5, 8. For every fixed u, we find that V(u; b) is a concave function of b, which means that V(u; b) first increases and then decreases in u. There are also two factors making contributions to this phenomenon. On one hand, smaller threshold level b means that more dividends can be paid off in finite time, but the surplus process is more prone to ruin due to claims. On the other hand, when the threshold level b is larger, the surplus process has fewer chances of going above this level, and consequently dividend payments would become small. Fig. 3 motivates us to find the optimal threshold level, say bopt , which maximizes the expected discounted dividend payments before ruin. In general, it is hard to find analytical expression for bopt . Then we get bopt by comparing values of V(u; b) for different threshold levels. In Table 1, we provide some values for bopt . It follows from these values that bopt depends on the initial surplus level u, which is different from the classical theory in the literature of dividend strategy. In Fig. 4, we compare V(u; b) for different claim

10

Z. Zhang, X. Han / Applied Mathematics and Computation 315 (2017) 1–12

a

b 40

35

c 35

35

30

30

25

25 V(u;b)

V(u;b)

V(u;b)

30

25

20

20 20 u=3 15

10

0

2

4

6

u=3

15

u=5

u=5

u=8

u=8

8

10

10

u=5 u=8 10

0

2

4

b

6

8

u=3

15

10

0

2

4

6

8

10

b

b

Fig. 3. The expected discounted dividends before ruin as a function of b for u = 3, 5, 8. (a) fY (y ) = 3e− 2 y − 3e−3y ; (b) fY (y ) = e−y ; (c) fY (y ) = 4 −2y e . 3 3

a

b

35

1 − 12 y e 6

+

c

40

40

35

35

30

30

30

V(u;b)

V(u;b)

V(u;b)

25 25

25

20 20

−3y/2

fY(y)=3e

15

20 f (y)=3e−3y/2−3e−3y

−3y

−3e

10 1

2

3

4

5 u

6

7

8

9

−y/2

f (y)=1/6e

Y

Y

0

−y

fY(y)=e

f (y)=1/6e−y/2+4/3e−2y

f (y)=1/6e−y/2+4/3e−2y 10

0

2

4

6

8

−3y

−3e

Y

15

−y

fY(y)=e

Y

10

−3y/2

f (y)=3e

Y

15

f (y)=e−y

Y

10

10

0

2

4

u

6

−2y

+4/3e

8

10

u

Fig. 4. The expected discounted dividends before ruin as a function of u for different claim size distributions. (a) b = 3; (b) b = 5; (c) b = 8.

Table 1 Some exact values of bopt when δ = 0.05, γ = 0.5, λ = 1, c = 2, α = 0.5.

fY (y ) = 3e− 2 y − 3e−3y fY (y ) = e−y y fY (y ) = 16 e− 2 + 43 e−2y 3

u=0

u=2

u=4

u=6

u=8

u = 10

u = 12

u = 14

u = 16

5.14 5.80 6.27

5.14 5.80 6.27

5.14 5.80 6.27

5.64 6.00 6.27

5.89 6.68 7.60

5.97 6.85 7.92

5.99 6.91 8.06

5.99 6.93 8.13

5.99 6.93 8.16

size densities, and find that V(u; b) is a decreasing function of the variance of the claim sizes. This is possibly due to that ruin is likely to happen earlier for claim sizes with larger variance. Next, we study the Gerber–Shiu function. Using the same parameters as above, we consider the Laplace transform of the ruin time, which is a special Gerber–Shiu function with penalty function w ≡ 1. In Fig. 5 we plot φ (u; b) as a function of u and study the impact of initial surplus on φ (u; b). It follows that φ (u; b) is a decreasing function of u, which is due to that the surplus process would stay at a high level for large initial surplus u, so that ruin is unlikely to happen and the ruin time τ b becomes large. Furthermore, Fig. 5 implies that φ (u; b) converges to a nonzero constant as u → ∞. This fact has also been observed by Zhang and Cheung [22] in the periodic dividend strategy. In Fig. 6, we plot the Laplace transform of ruin time as a function of the threshold level b, and we find that φ (u; b) is a decreasing function of b for each fixed initial surplus u. We can explain this phenomenon as follows. When the threshold level b becomes larger, fewer dividends would be paid off so that the surplus process is more likely to stay at a high level and ruin is unlikely to happen in this situation. Finally, in Fig. 7 we compare φ (u; b) for different claim size densities. We find that the larger the variance of the claim size, the larger φ (u; b). As is remarked as in the last paragraph, this is possibly due to that claim sizes with large variance may cause higher risk.

Z. Zhang, X. Han / Applied Mathematics and Computation 315 (2017) 1–12

b

a

c 0.7

0.7

0.8

b=3

b=3 0.6

b=5

b=3

b=5

0.6

b=8

0.5

0.6

0.4

0.4

0.5 φ(u;b)

φ(u;b)

φ(u;b)

0.5

0.3

0.3

0.4

0.2

0.2

0.3

0.1

0.1

0.2

0

2

4

6

b=5

0.7

b=8

b=8

0

11

8

0

10

0.1

0

2

4

6

u

8

10

0

1

2

3

4

5 u

u

6

Fig. 5. The Laplace transform of the ruin time as a function of u for b = 3, 5, 8. (a) fY (y ) = 3e− 2 y − 3e−3y ; (b) fY (y ) = e−y ; (c) fY (y ) = 3

b

a 0.9

0.6

0.6

0.6

0.5

0.5

0.5

φ(u;b)

0.7

φ(u;b)

φ(u;b)

0.7

0.4

0.4

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0

0

6

8

10

0

1

2

3

4

b

u=5 u=8

5 b

6

7

8

9

0

10

0

2

4

6

3

b 0.7 fY(y)=3e

0.65

−3e

−3y/2

fY(y)=3e

−y −y/2

Y

0.6

+

4 −2y e . 3

f (y)=3e−3y/2−3e−3y

−3y

−3e

Y

f (y)=e−y

0.6

f (y)=1/6e

1 − 12 y e 6

0.7

−3y

fY(y)=e

10

c 0.7

−3y/2

8

b

Fig. 6. The Laplace transform of the ruin time as a function of b for u = 3, 5, 8. (a) fY (y ) = 3e− 2 y − 3e−3y ; (b) fY (y ) = e−y ; (c) fY (y ) =

a

4 −2y e . 3

0.4

0.3

4

10

u=3 0.8

u=5 u=8

u=8

2

+

u=3 0.8

u=5

0

1 − 12 y e 6

9

0.9

u=3

0.7

8

c 0.9

0.8

7

−2y

+4/3e

−y

fY(y)=e

0.6

Y

−y/2

f (y)=1/6e Y

fY(y)=1/6e−y/2+4/3e−2y

−2y

+4/3e

0.5

0.5

0.4

0.4

phi(u;b)

φ(u;b)

φ(u;b)

0.55

0.5

0.3

0.3

0.2

0.2

0.1

0.1

0.45

0.4

0.35

0

2

4

6 u

8

10

0

0

2

4

6 u

8

10

0

0

2

4

6

8

10

u

Fig. 7. The Laplace transform of the ruin time as a function of u for different claim size distributions. (a) b = 3; (b) b = 5; (c) b = 8.

5. Concluding remarks In the context of the classical compound Poisson risk model, we have extended the threshold dividend strategy to a mixed dividend strategy. We derive integro-differential equations for the expected discounted dividend payments before ruin and the Gerber–Shiu discounted penalty function. Analytical solution to the equations is derived and numerical examples are provided to illustrate the theoretical results. In this paper, we have assumed that solvency is continuously checked, i.e. ruin is declared immediately when the insurer’s surplus level becomes negative. As in [2,3], we can also consider the case when the solvency is discretely monitored. On the other hand, we have assumed that the inter-dividend-decision times follow exponential distribution. This assumption can also be extended to the Erlang arrival assumption, but under the later assumption the derivation procedure would

12

Z. Zhang, X. Han / Applied Mathematics and Computation 315 (2017) 1–12

be very involved. Finally, we would like to remark that the randomized observation strategy can also be used in epidemic systems. We leave this as an open problem and refer the interested readers to the related papers [10,11,16–18]. Acknowledgments The authors would like to thank the two anonymous referees for helpful comments and suggestions which have improved an earlier version of the paper. Zhimin Zhang is supported by the National Natural Science Foundation of China (grant nos. 11471058, 11661074) and MOE (Ministry of Education in China) Project of Humanities and Social Sciences (16YJC910 0 05). References [1] H. Albrecher, J. Hartinger, S. Thonhauser, On exact solutions for dividend strategies of threshold and linear barrier type in a Sparre Andersen model, ASTIN Bull. 37 (2007) 203–233. [2] H. Albrecher, E.C.K. Cheung, S. Thonhauser, Randomized observation periods for the compound Poisson risk model: dividends, ASTIN Bull. 41 (2011) 645–672. [3] H. Albrecher, E.C.K. Cheung, S. Thonhauser, Randomized observation periods for the compound Poisson risk model: the discounted penalty function, Scand. Actuar. J. 6 (2013) 424–452. [4] B. Avanzi, E.C.K. Cheung, B. Wong, J.-K. Woo, On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency, Insur. Math. Econ. 52 (2013) 98–113. [5] X.J. Chen, A. Szolnoki, M. Perc, Averting group failures in collective-risk social dilemmas, EPL 99 (2012) 68003. [6] X.J. Chen, Y.L. Zhang, T.Z. Huang, M. Perc, Solving the collective-risk social dilemma with risky assets in well-mixed and structured populations, Phys. Rev. E 90 (2014) 052823. [7] M.C.H. Choi, E.C.K. Cheung, On the expected discounted dividends in the Cramér-Lundberg risk model with more frequent ruin monitoring than dividend decisions, Insur. Math. Econ. 59 (2014) 121–132. [8] D.C.M. Dickson, C. Hipp, On the time to ruin for Erlang(2) risk processes, Insur. Math. Econ. 29 (2001) 333–344. [9] H.U. Gerber, E.S.W. Shiu, On the time value of ruin, N. Am. Actuar. J. 2 (1998) 48–78. [10] L. Li, Bifurcation and chaos in a discrete physiological control system, Appl. Math. Comput. 252 (2015) 397–404. [11] L. Li, Monthly periodic outbreak of hemorrhagic fever with renal syndrome in china, J. Biol. Syst. 24 (2016) 519–533. [12] S. Li, J. Garrido, On ruin for the Eralng(n) risk model, Insur. Math. Econ. 34 (2004) 391–408. [13] X.S. Lin, K.P. Pavlova, The compound Poisson risk model with a threshold dividend strategy, Insur. Math. Econ. 38 (2006) 57–80. [14] M. Perc, A. Szolnoki, Coevolutionary games-a mini review, BioSystems 99 (2010) 109–125. [15] Y. Shimizu, Z.M. Zhang, Estimating Gerber-Shiu functions from discretely observed levy driven surplus, Insur. Math. Econ. 74 (2017) 84–98. [16] G.Q. Sun, C.H. Wang, Z.Y. Wu, Pattern dynamics of a Gierer-Meinhardt model with spatial effects, Nonlinear Dyn. 88 (2017) 1385–1396. [17] G.Q. Sun, J.H. Xie, S.H. Huang, Z. Jin, M.T. Li, L.Q. Liu, Transmission dynamics of cholera: mathematical modeling and control strategies, Commun. Nonlinear Sci. 45 (2017) 235–244. [18] G.Q. Sun, Z. Zhang, Global stability for a sheep brucellosis model with immigration, Appl. Math. Comput. 246 (2014) 336–345. [19] N. Wan, Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion, Insur. Math. Econ. 40 (2007) 509–523. [20] Z. Wang, L. Wang, A. Szolnoki, M. Perc, Evolutionary games on multilayer networks: a colloquium, Eur. Phys. J. B 88 (2015) 124. [21] Z.M. Zhang, On a risk model with randomized dividend-decision times, J. Ind. Manage. Optim. 10 (2013) 1041–1058. [22] Z.M. Zhang, E.C.K. Cheung, The Markov additive risk process under an Erlangized dividend barrier strategy, Methodol. Comput. Appl. Probab. 18 (2016) 275–306. [23] Z.M. Zhang, E.C.K. Cheung, H.L. Yang, Lévy insurance risk process with Poissonian taxation, Scand. Actuar. J. 2017 (2017) 51–87. [24] Z.M. Zhang, E.C.K. Cheung, H.L. Yang, On the compound Poisson risk model with periodic capital injections, ASTIN Bull., 2017. in press. [25] Z.M. Zhang, C.L. Liu, Moments of discounted dividend payments in a risk model with randomized dividend-decision times, Front. Math. China 12 (2017) 493–513. [26] Z.M. Zhang, Y. Yang, C.L. Liu, On a perturbed compound Poisson model with varying premium rates, J. Ind. Manage. Optim. 13 (2017) 721–736.